Rationalising the Denominator

Multiply by a fraction equal to one that removes the square root from the denominator.

1. Multiplying by \dfrac{\sqrt{11}}{\sqrt{11}} leaves the value unchanged because this fraction equals 1.
2. The new denominator is \sqrt{11}\times\sqrt{11}=11, which is rational.

Multiply the numerator and denominator by the square root in the denominator.

1. Multiply the numerator and denominator by \sqrt{13}. This is multiplying by \frac{\sqrt{13}}{\sqrt{13}}=1, so the value does not change.
2. The numerator becomes 1\times\sqrt{13}=\sqrt{13} and the denominator becomes \sqrt{13}\times\sqrt{13}=13.
3. So \frac{1}{\sqrt{13}}=\frac{\sqrt{13}}{13}.

After rationalising, cancel any common factor.

1. Multiply top and bottom by \sqrt2 to get \dfrac{8\sqrt2}{2}.
2. Cancel the factor 2 to get 4\sqrt2.

Multiply both the numerator and denominator by the same square root.

1. Multiply the numerator and denominator by \sqrt5, which keeps the value the same.
2. The numerator becomes 3\sqrt5 and the denominator becomes \sqrt5\times\sqrt5=5.
3. So the answer is \frac{3\sqrt5}{5}; 3 and 5 have no common factor, so it cannot be simplified.

Keep the numerical factor in the denominator when you multiply by the square root.

1. Multiply top and bottom by \sqrt7: the fraction becomes \dfrac{7\sqrt7}{14}.
2. Cancel the factor 7 to get \dfrac{\sqrt7}{2}.

Simplify the square root before rationalising.

1. Since \sqrt{27}=3\sqrt3, the fraction is \dfrac4{\sqrt3}.
2. Multiply top and bottom by \sqrt3 to get \dfrac{4\sqrt3}{3}.

Multiply the two square roots in the numerator and then simplify the resulting surd.

1. Multiply top and bottom by \sqrt6 to get \dfrac{\sqrt{60}}6.
2. Since \sqrt{60}=2\sqrt{15}, the answer simplifies to \dfrac{\sqrt{15}}3.

Multiply every term in the numerator by the rationalising factor.

1. Multiply the whole numerator and denominator by \sqrt6.
2. This gives \dfrac{5\sqrt6+6}{6}.
3. Dividing both numerator terms by 6 gives 1+\dfrac{5\sqrt6}{6}.

Collect like surds in the numerator first.

1. Since \sqrt{20}=2\sqrt5, the numerator simplifies to \sqrt5.
2. Multiply \dfrac{\sqrt5}{\sqrt{10}} by \dfrac{\sqrt{10}}{\sqrt{10}} to get \dfrac{\sqrt{50}}{10}.
3. Using \sqrt{50}=5\sqrt2, the answer is \dfrac{\sqrt2}{2}.

Rationalise before cancelling the numerical factor.

1. Multiply top and bottom by \sqrt6 to get \dfrac{\sqrt{18}}{24}.
2. Since \sqrt{18}=3\sqrt2, cancel the factor 3.
3. The answer is \dfrac{\sqrt2}{8}.

Divide by the coefficient of the unknown, then rationalise.

1. Divide by \sqrt8 to get x=\dfrac6{\sqrt8}.
2. Since \sqrt8=2\sqrt2, this is \dfrac3{\sqrt2}.
3. Rationalising gives x=\dfrac{3\sqrt2}{2}.

Divide the area by the length to find the width.

1. The width is \dfrac{48}{4\sqrt6}=\dfrac{12}{\sqrt6} cm.
2. Multiply top and bottom by \sqrt6 to get \dfrac{12\sqrt6}{6}.
3. The width is 2\sqrt6 cm.

Find the scale factor from the second part of the ratio.

1. The ratio u:v=2:\sqrt3 means \frac{u}{v}=\frac{2}{\sqrt3}, so u=\frac{2}{\sqrt3}\times v.
2. With v=5: u=\frac{2\times5}{\sqrt3}=\frac{10}{\sqrt3}.
3. Multiply the numerator and denominator by \sqrt3: u=\frac{10\sqrt3}{3}.

Undo the division by the square root on the left.

1. Multiply both sides by \sqrt5.
2. This gives k=\dfrac{3\sqrt5\times\sqrt5}{2}=\dfrac{15}{2}.

Simplify the second denominator before combining the fractions.

1. Simplify the second denominator: \sqrt{12}=\sqrt{4\times3}=2\sqrt3, so \frac{2}{\sqrt{12}}=\frac{2}{2\sqrt3}=\frac{1}{\sqrt3}.
2. The fractions now have the same denominator, so add the numerators: \frac{5}{\sqrt3}+\frac{1}{\sqrt3}=\frac{6}{\sqrt3}.
3. Multiply the numerator and denominator by \sqrt3: \frac{6\sqrt3}{3}.
4. Cancel the 3: the answer is 2\sqrt3.

Rationalise the whole fraction, then simplify each surd in its numerator.

1. Multiply top and bottom by \sqrt{10}.
2. The fraction becomes \dfrac{\sqrt{60}+\sqrt{150}}{10}.
3. Using \sqrt{60}=2\sqrt{15} and \sqrt{150}=5\sqrt6, the answer is \dfrac{2\sqrt{15}+5\sqrt6}{10}.

Find the reciprocal, then write both terms with rational denominators before adding.

1. Rationalising gives t=\dfrac{3\sqrt5}{5}.
2. The reciprocal of the given value is \dfrac1t=\dfrac{\sqrt5}{3}.
3. Use the common denominator 15: t+\dfrac1t=\dfrac{9\sqrt5+5\sqrt5}{15}=\dfrac{14\sqrt5}{15}.

Divide by the surd coefficient and multiply the whole numerator by the rationalising factor.

1. Since \sqrt{12}=2\sqrt3, x=\dfrac{9-\sqrt3}{2\sqrt3}.
2. Multiply top and bottom by \sqrt3 to get \dfrac{9\sqrt3-3}{6}.
3. Cancel the factor 3 to get x=\dfrac{3\sqrt3-1}{2}.

Dividing by a fraction is the same as multiplying by its reciprocal.

1. Dividing by a fraction is multiplying by its reciprocal: \frac{a}{b}=\frac{3}{\sqrt2}\times\frac{\sqrt6}{5}=\frac{3\sqrt6}{5\sqrt2}.
2. Since \frac{\sqrt6}{\sqrt2}=\sqrt{\frac{6}{2}}=\sqrt3, the surds simplify straight away.
3. So \frac{a}{b}=\frac{3\sqrt3}{5}, which already has a rational denominator.

Expand the squared numerator and simplify the denominator before rationalising.

1. Expand the numerator to get 11+6\sqrt2.
2. Since \sqrt8=2\sqrt2, the fraction is \dfrac{11+6\sqrt2}{2\sqrt2}.
3. Multiply top and bottom by \sqrt2 to get \dfrac{11\sqrt2+12}{4}.
4. The answer is 3+\dfrac{11\sqrt2}{4}.